Generalized block anti-Gauss quadrature rules

Golub and Meurant describe how pairs of Gauss and Gauss–Radau quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real-valued matrix functionals defined by a symmetric matrix. However, there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds. In this situation, it may be possible to determine upper and lower bounds by evaluating pairs of Gauss and anti-Gauss rules. Unfortunately, it is difficult to ascertain whether the values determined by Gauss and anti-Gauss rules bracket the value of the given real-valued matrix functional. Therefore, generalizations of anti-Gauss rules have recently been described, such that pairs of Gauss and generalized anti-Gauss rules may determine upper and lower bounds for real-valued matrix functionals also when pairs of Gauss and (standard) anti-Gauss rules do not. The available generalization requires the matrix that defines the functional to be real and symmetric. The present paper reviews available anti-Gauss and generalized anti-Gauss rules and extends them in several ways that allow applications in new situations. In particular, the genarlized anti-Gauss rules for a real-valued non-negative measure described in Pranić and Reichel (J Comput Appl Math 284:235–243, 2015 ) are extended to allow the estimation of the error in matrix functionals defined by a non-symmetric matrix, as well as to matrix-valued matrix functions. Modifications that give simpler formulas and thereby make the application of the rules both easier and applicable to a larger class of problems also are described.